(O Level Phy) The simple pendulum explained

The first chapter of an O Level physics course is typically about physical quantities. Usually, this chapter ends with the section on time measurements which includes a practical session with the simple pendulum. This experiment is challenging in every way from understanding its concepts to executing it correctly. It is like the crash course in the first few months of a national service (army life). Yet, it is important in exposing you to the techniques of plotting good graphs.

Especially if you take the pure physics course, there will be an unfamiliar equation thrown into your face. This formula relates the period T (time taken for the pendulum to execute one oscillation or swing) to the length L of the pendulum (distance from the fixed point of suspension to the centre of the bob). 

T = 2π√(L/g)

How on Earth is this formula obtained? Some formulas especially those from laws like Newton's Second Law, Hooke's Law and Ohm's Law are obtained empirically or experimentally. Some like this one that you are looking at are obtained through mathematical derivation. To derive it, scientists make some assumptions such as that the angle at which the pendulum bob is displaced is so small that the bob moves to and fro along a straight line. That is why you should not swing the pendulum from a large angle of displacement and should make sure that it is not swinging in elliptical path. (FYI: The to-and-fro motion of the pendulum is called Simple Harmonic Motion SHM which is learned in A Level Physics. I've written another example of SHM here.) 

Swing from a small angle and make sure the swing is not elliptical... 

The aim of the experiment is often written as "To find the acceleration due to gravity, g". (Here, g is not grams.) "What's that?” you may ask since you have never come across the term acceleration due to gravity which will be explained to you in later chapters. For now, take it as the rate of increase in speed of a falling object. "But the pendulum bob is swinging, not dropping." Well, it swings because of the force of gravity pulling the bob.

To impart the skill of finding gradient of the straight line graph, you are asked to plot a graph of T² against L. Why T² against L and not T against L? If you try to plot T against L, you will get a curve, not a straight-line graph.

Find the gradient of the graph of T² against L... 

By plotting T² against L, you can then calculate the gradient of the straight-line graph to find the acceleration due to gravity, g.

But how do we know it is T² against L and not something like T against L²? Again, it's from math. Now, it is the time you get used to that science and math are inseparable. We know we have to plot T² against L by squaring both sides of the simple pendulum equation.


T² = 4π²L/g



Comparing this to the general equation for the straight line graph y = mx + c, we see that T² is y, 4π²/g is m and L is x. The y-intercept c is 0 and so you will expect to get a straight line graph that passes through the origin if you plot T² against L. Your additional math teacher will teach this skill under the chapter on straight line graphs.

Alternatively, you can plot T against the square root of L to get a straight line graph but the gradient will be equal to 2π/√g.

Or find the gradient of T against √L.